In a time-of-flight mass spectrometer (which is hereinafter abbreviated to TOFMS), the time of flight required for ions ejected from an ion source to reach an ion detector is measured, and the mass (or mass-to-charge ratio m/z, to be exact) of each ion is calculated from the time of flight of that ion. One major cause of deterioration in the mass-resolving power is the initial energy distribution of the ions. An initial energy distribution of the ions ejected from the ions source causes a time-of-flight distribution of the ions of the same mass and deteriorates the mass-resolving power. To compensate for the time-of-flight distribution due to the initial energy distribution of the ions, ion reflectors have been widely used. A TOFMS using an ion reflector is hereinafter called the “reflectron” according to the common practice.
An ion reflector has an electric potential increasing in the traveling direction of the ions, and has the function of repelling ions coming through a field-free drift space. An ion having a higher initial energy (initial speed) penetrates deeper into the ion reflector and hence spends a longer period of time flying in the ion reflector when reflected. On the other hand, an ion having a larger amount of initial energy flies at a higher speed and hence spends a shorter period of time flying through the drift space of a fixed length. Therefore, by appropriately adjusting the parameters so as to cancel the increase in the time of flight in the ion reflector by the decrease in the time of flight in the drift space, it is possible to create a state in which the total time of flight from the ion source to the detector is almost independent of the initial energy within a certain range of energy (see Non-Patent Document 1 for details). Such an operation of focusing the same kind of ions with different amounts of kinetic energy on the time-of-flight axis so as to make them simultaneously arrive at the detector is hereinafter called the “energy focusing” according to the common practice.
To date, various types of reflectrons have been developed. They can be roughly divided into two groups: one type is a multi-stage system in which a plurality of regions with a uniform (or nearly uniform) electric field are connected in series, and the other type is a non-uniform electric field system in which the potential continuously changes with the intensity of the electric field defined as a function of the distance. Initially, the multi-stage system is hereinafter described.
The structurally simplest version of the multi-stage system is the single-stage reflectron. A potential of the single-stage reflectron is schematically shown in FIG. 23 (see Non-Patent Document 1). The ion reflector has a uniform electric field (i.e. the potential U is proportional to the distance X). A grid electrode G through which ions can pass is provided at the boundary between the field-free drift region and the ion reflector. In the figure, X=0 is the location of the flight start point and the detection point of the ions, L is the length of the field-free drift space, and a is the penetration depth of the ions into the ion reflector. In this system, if the initial energy of an ion satisfies the following equation (1), the time-of-flight distribution is compensated for up to the first derivative of the energy, and as a result, the first-order energy focusing (which is hereinafter simply called the “first-order focusing”) is achieved:L=2·a  (1)However, in the case of the first-order focusing, since the compensation for the time-of-flight distribution is not achieved for the second and higher-order derivatives of the energy, a high mass-resolving power can be achieved only for ions having a comparatively narrow energy distribution. In the following description, the position corresponding to the depth a in the single-stage reflectron is called the “first-order focusing position.”
FIG. 24 is a schematic potential diagram of a dual-stage reflectron. The dual-stage reflectron was developed for the first time by Mamyrin et al. (see Non-Patent Document 2). As shown in FIG. 24, the ion reflector consists of two uniform electric fields, with a grid electrode G provided as a partition at the boundary between the field-free drift region and the first uniform electric field (first stage) as well as at the boundary between the first uniform electric field and the second uniform electric field (second stage). If the first stage is adequately short, and if approximately two thirds of the initial energy is lost in the first stage, the time-of-flight distribution is compensated for up to the second derivative of the energy. That is to say, the second-order energy focusing (which hereinafter is simply called “the second-order focusing”) is achieved and a high mass-resolving power is obtained.
According to an analysis by Boesl et al., exact conditions to be satisfied for achieving the second-order focusing in a dual-stage reflectron are given by the following equations (2) (See Non-Patent Document 3. It should be noted that the equations given in the original paper are incorrect; the following equations (2) are the recalculated ones):a=[(c−2b)/2(b+c)]·{b+[√{square root over (3)}·(c−2b)3/2/9√{square root over (c)}]}p=2(b+c)/3c  (2)where a is the penetration depth of the ion into the second stage, b is the length of the first stage, c is the length of the field-free drift region, and p is the proportion of the ion energy to be lost in the first stage. Equations (2) suggest that, if the lengths b and c are given, the values of a and p which satisfy the second-order focusing condition can be uniquely determined. In the dual-stage reflectron, since the time-of-flight distribution is compensated for up to the second derivative of the ion energy, a high mass-resolving power can be achieved for ions having a broader energy distribution than in the case of the single-stage reflectron. In the following description, the position corresponding to the depth a in the dual-stage reflectron is called the “second-order focusing position.”
As for a multi-stage reflectron, which can be conceived as an extension of the dual-stage reflectron, it can be generally expected that increasing the number of uniform electric fields (or nearly uniform electric fields) in the multi-stage reflectron improves the performance, with the time-of-flight distribution being compensated for up to higher-order derivatives of the ion energy (the cancellation of up to the N-th derivative is hereinafter called the “N-th order focusing”), thus making it possible to achieve a high mass-resolving power for ions having a broader energy distribution. The possibility of improving the performance for an actual increase in the number of stages has been studied by numerical computations in Non-Patent Document 5, which includes a report of the results obtained by correcting the design parameters up to higher-order derivatives without departing from the practically acceptable ranges of the parameters while increasing the number of stages in the ion reflector up to four. However, increasing the number of stages does not lead to a significant increase in the energy range in which a high level of mass-resolving power is obtained. Furthermore, increasing the number of grid electrodes placed on the flight path of the ions causes a greater amount of loss of the ions and deteriorates the sensitivity. Such a system can be said to be practically unusable.
In view of such limits of the multi-stage reflectron, the non-uniform electric field system has been developed as an attempt to reduce the time-of-flight distribution of the ions having an even broader energy distribution. An ideal pinnacle of this system is a reflectron using a simple harmonic motion.
That is to say, as can be understood from the motion of a weight attached to an end of a spring, when the potential U is given by a harmonic function expressed by the following equation (3), the time of flight (TOF) of an ion will be equal to one half of the period of the simple harmonic motion, as given by the following equation (4):U=(½)·k·X2  (3)TOF=π√{square root over (m/k)}  (4)where m is the mass of the ion, and k is a constant.
These equations demonstrate that the time of flight is independent of the initial energy, and isochronism is exactly achieved. However, in practice, the absence of the field-free drift region in the potential distribution, as in the case of the harmonic function of equation (3), is a considerably serious drawback for TOFMS, because, without a field-free drift region, the ion source and the detector cannot be placed at any position other than at the bottom of the potential, which imposes an extremely strong restriction on the system design. A technique for solving this drawback is disclosed in Patent Document 1 and Non-Patent Document 4, in which the sum of a potential proportional to the distance X and a potential proportional to the square of the distance X is used as the potential inside the reflector, with the aim of reducing the time-of-flight distribution even in the case where a field-free drift region is connected to an ion reflector having a gradient electric field. This method ensures a certain level of energy-focusing performance over a comparatively broad range of energy. However, it also makes the negative effect of breaking the exact isochronism, thus limiting the improvement of the mass-resolving power.
On the other hand, a configuration of a TOF-TOF system for performing an MS2 analysis is described in Patent Document 2. In this system, a non-uniform electric-field potential is created inside a second ion reflector for the purpose of the energy focusing of the fragment ions produced in a collision cell. In another ion reflector described in Patent Document 4 and Non-Patent Document 7, the entire ion reflector is divided into a decelerating region as the first section and a (non-uniform) correcting potential region as the second section. These documents demonstrate that applying an appropriate non-uniform electric field on the correcting potential region makes the time of flight of ions completely independent of their initial energy (equal to or higher than a specific threshold) over the entire time-of-flight range, i.e. that complete isochronism is theoretically achievable. Specifically, it is shown that an ideal (one-dimensional) potential distribution on the central axis of the correcting potential region can be determined by integral equations. One example is also presented, in which the result of integration is expressed as an analytical function form.
A system which has significantly contributed to the practical realization of a reflectron having both high mass-resolving power and high energy-focusing performance (i.e. high sensitivity) is the system described in Patent Document 3. This system, which can be regarded as a compromise between the multi-stage uniform electric-field system and the non-uniform electric-field system, is a variation of the dual-stage reflectron in which the entire first stage and a portion of the second stage are designed to be a uniform electric-field region using an approximately constant electric field, while the remaining portion extending to the end is designed to be a correcting potential region with a non-uniform electric field adopted therein, whereby the electric-field strength on the central axis is made to substantially increase. To avoid the loss of the ions, no grid electrode is used. The electric-field strength in the first stage is set at a low level so as to improve the ion-beam focusing performance, with a corresponding sacrifice of the mass-resolving power. The amount of correction of the electric-field strength in the second stage is 10% of the uniform electric-field strength or even smaller. However, according to the document, since the equipotential surfaces in the grid-less reflector are not flat but curved, the trajectory of an ion traveling on a path dislocated from the central axis will diverge due to the lens effect. Nevertheless, an advantage still exists in that a higher level of mass-resolving power can be obtained for ions having a broader energy distribution than before, and the system has already been put into practice.
Based on the previously described past improvements of the reflectron, an ideal reflectron is herein defined as “a reflectron capable of the energy focusing up to infinitely high-order terms for a time-of-flight distribution by using a potential distribution created by a non-uniform electric field, at any energy level equal to or higher than a specific level E0.” As will be described later, the following five basic conditions must be satisfied for the practical realization of an ideal reflectron.
<1: Complete Isochronism> It should be possible to achieve energy focusing up to infinitely high-order terms with respect to the time of flight.
<2: Suppression of Beam Divergence> A divergence of the beam in the reflector should be suppressed.
<3: Suppression of Off-Axis Aberration> An off-axis aberration, i.e. the temporal aberration for an ion on a path dislocated from the central axis, should be suppressed.
<4: Feasibility of Potential> It should be possible to produce an ideal potential in a practical way, using a limited number of electrodes.
<5: Tolerance for Non-Uniform Electric Field before Correction> As will be described later, it should be possible to realize a practically usable ideal potential even if a non-uniform electric field is present in the vicinity of the beginning portion of the correcting potential before the correction.
The condition <1: Complete Isochronism> can be expressed by the following equation (5):T(E)=T(E0)+(dT/dE)(E−E0)+(½)(d2T/dE2)·(E−E0)2+(⅙)(d3T/dE3)(E−E0)3+ . . .  (5),where E is the initial energy of the ion, and T(E) is the time of flight of the ion.
A Wiley-McLaren solution, as already described, uses the first-order focusing which cancels the terms of equation (5) up to the first-order differential coefficient by a single-stage reflectron, using a potential created by a uniform electric field. A Mamyrin solution uses the second-order focusing which cancels the terms of equation (5) up to the second-order differential coefficient by a dual-stage reflectron. These solutions cannot be regarded as an ideal reflectron, because the former system leaves the second and higher-order differential coefficients intact, while the latter system leaves the third and higher-order differential coefficients intact.
The conditions <2: Suppression of Beam Divergence> and <3: Suppression of Off-Axis Aberration> are also essential for the practical realization of an ideal reflectron. Both the beam dispersion and the temporal aberration occur due to the fact that divE≠0 for a non-uniform electric field in vacuum. Firstly, if the discrepancy from the uniform electric field is large or the curvature of the potential distribution is large, the ion reflector acts as a concave lens, which causes a divergence of the ion trajectory and eventually lowers the signal intensity. Secondly, even if an ideal potential along the central axis is realized, a potential discrepancy inevitably occurs for a trajectory dislocated from the central axis, which causes a temporal aberration and eventually lowers the mass-resolving power. In the following description, the former problem is called the “divergence problem” and the latter is called the “temporal aberration due to the off-axis location.”
The condition <4: Feasibility of Potential> is also important from practical points of view, because, even if a correcting potential to be created inside the reflector to achieve complete isochronism has been theoretically determined, that potential cannot always be actually created as a three-dimensional potential distribution. In other words, even if a one-dimensional potential distribution having ideal values on the central axis (which is hereinafter called the “1D-IDL”) has been found, it is not guaranteed that a three-dimensional potential distribution which has been simulated based on the 1D-IDL (the simulated distribution is hereinafter called the “3D-SIM”) is a practical approximation of the 1D-IDL, because there is the absolute restriction that 3D-SIM should be a solution of the Laplace equation. A strong, specific concern is that an ideal correcting potential has a specific characteristic (which will be described later) at the starting point of the correcting potential, i.e. that the high-order differential coefficients relating to the position of the correcting potential inevitably diverge. As a result, under an electrostatic constraint, the correcting potential can merely be reproduced as an approximation. Accordingly, for the practical realization of an ideal reflectron, it is essential to determine whether a practical isochronism can be achieved by an approximate potential distribution created by a limited number of guard-ring electrodes. In the following description, this determination task is mainly substituted for by the task of initially determining a 1D-IDL, which can be obtained with almost zero numerical discrepancy from ideal values, and then performing numerical calculations on a large scale to obtain a three-dimensional approximate solution 3D-SIM corresponding to that 1D-IDL for a specified set of electrodes.
As will be described later, in the conventional theory for the ideal reflectron, the correcting potential is analytically determined on the premise that the electric field which serves as a base before the correction in the vicinity of the beginning portion of the correcting potential is a uniform electric field. However, a study by the present inventors has revealed that the electric field at the grid electrode placed at the boundary of the electric field is actually disordered due to the seeping of the electric field or other factors, and this disorder fatally deteriorates the isochronism. In the case of a grid-less reflector with no grid electrode, the problem is even more serious because the degree of non-uniformity of the electric field is greater. Accordingly, for the practical realization of an ideal reflectron, the correcting potential must be obtained by using a system as close to a real form as possible. Thus, it is necessary to satisfy the condition <5: Tolerance for Non-Uniform Electric Field before Correction>, which requires that the ideal potential should be applicable even in the case where the electric field which serves as a base in the vicinity of the beginning portion of the correcting potential before the correction is a non-uniform electric field.
Now, let the previously described conventional techniques be evaluated from the viewpoints of the five basic conditions. The <1: Complete Isochronism> has already been achieved, for example, in Patent Document 4 and Non-Patent Document 7 (which are hereinafter called “the documents of Cotter et al.”) That is to say, a general solution for the potential distribution of an ideal reflectron capable of the energy focusing up to infinitely high orders has already been obtained in those documents. However, the solution described in those documents is limited to a one-dimensional space (with ions moving on the central axis); there is no mention of how to satisfy the basic conditions relating to three-dimensional motions, such as the conditions <2: Suppression of Beam Divergence> and <3: Suppression of Off-Axis Aberration>. Therefore, no ideal reflectron which can achieve both high mass-resolving power and high sensitivity has yet been practically realized. That is to say, at least either the mass-resolving power or the sensitivity is sacrificed in the currently used reflectrons.
The system described in Patent Document 3 does not exactly satisfy the condition <1: Complete Isochronism>. However, as compared to the reflectrons known by that time, it has achieved a higher level of mass-resolving power for ions with a broader energy distribution. In this respect, the system can be said to be closer to an ideal reflectron. However, a problem exists in that it requires repeating a trial and error process in a computer simulation in order to find a sufficient potential distribution for achieving a required mass-resolving power. It is impractical to use such a trial and error process in order to reach an ideal extremity, i.e. a solution that exactly satisfies the conditions for isochronism. The energy range in which a practical isochronism is achieved is also limited.
The technique described in the documents of Cotter et al. takes the following steps to practically realize an ideal potential:
[Step 1] An ideal potential distribution in the correcting potential region is expressed as a general solution including design parameters (distance and voltage).
[Step 2] The general solution obtained in Step 1 is expanded into a half-integer power series of (U−E0).
[Step 3] The design parameters are adjusted so that the expansion coefficients obtained in Step 2 will be individually zeroed.
However, taking the aforementioned steps is actually difficult, and in the first place, it is not always evident whether a solution which makes the coefficients in the power series expansion equal to zero actually exists. Furthermore, as already noted, having a general solution of the ideal potential is not enough for practical purposes; it does not make sense if no particular solution which satisfies the conditions <2: Suppression of Beam Divergence> through <5: Tolerance for Non-Uniform Electric Field before Correction> for embodying an ideal reflectron into an actual system is determined. Although Cotter et al. noted the general request that using an ideal one-dimensional potential distribution which has a smaller curvature and is closer to a straight line makes it easier to create an actual system, they proposed no specific technique for achieving that goal. More importantly, even if the method of Cotter et al. is used, it is possible that no practical solution with a small curvature of the potential distribution can be found for some setting of the design parameters. Actually, Cotter et al. mentions no solutions other than the first-order focusing solution, whose existence has already been proved to be evident.
A study by the present inventors has also revealed that, if in the first place there is no position where the N-th order focusing is achieved (this position hereinafter is simply called the “N-th order focusing position”), it is allowed to automatically conclude that there is no practical solution with a small curvature of the potential distribution. The N-th order focusing position can be more specifically defined as follows. Provided that the total time of flight is expressed as a function of energy E, the N-th order focusing position is the position on the central axis at which the potential value is equal to the energy E at which the first through N-th order derivative values are equal to zero. According to the studies on multi-stage reflectrons described in Patent Document 5 and other documents, the N-th order focusing position does not always exist in an arbitrary design; the fact is that there are considerable ranges of design parameters in which no N-th order focusing position can be found. This means that the situation with no N-th order focusing position existing from the start may more frequently occur depending on the setting of the design parameters.
Non-Patent Document 8 (which is hereinafter called the “document of Doroshenko”) is a study that succeeded the technique described in the documents of Cotter et al. Similar to Cotter et al., the study is focused on the one-dimensional model. In the documents of Cotter et al., the entire flight path of the ions including the ion source (or ion-accelerating region) is divided into a forward path (upstream region), a return path (downstream region) and a reflector region with a correcting potential, and a generalized integral equation for determining an ideal potential distribution within the reflector region for achieving isochronism for an arbitrary potential distribution on the forward and return paths is presented. On the other hand, in the document of Doroshenko, after an analogy between the reflection of ions by the reflector and the extraction of ions from the ion source is explained, a generalized integral equation for determining an ideal potential distribution inside the ion source for achieving isochronism in the extraction of the ions is described. A particular solution is also discussed, for the reason that using an ideal one-dimensional potential distribution whose curvature is smaller and closer to a straight line makes it easier to realize or design an actual system. In that discussion, on the premise that only a uniform electric field exists in the vicinity of the beginning portion of the correcting potential before the correction, Doroshenko demonstrated that the ideal correcting potential can be expanded into a half-integer power series of (U−E0), and that the curvature of the correcting potential can be kept small by achieving the first or second-order focusing. However, the premise that “only a uniform electric field exists in the vicinity of the beginning portion of the correcting potential before the correction” contradicts the aforementioned condition <5: Tolerance for Non-Uniform Electric Field before Correction>. In this respect, the problem for the practical realization is not solved at all.
As described thus far, although an ideal one-dimensional potential distribution has been obtained by conventional research and development efforts, no ideal reflectron has yet been realized. This is because no conventional technique can completely satisfy the basic conditions <1: Complete Isochronism> through <5: Tolerance for Non-Uniform Electric Field before Correction>. Practical realization of a three-dimensional, highly feasible, ideal reflectron which completely satisfies the conditions <1: Complete Isochronism> through <5: Tolerance for Non-Uniform Electric Field before Correction> has been strongly demanded for the purpose of improving the performance of mass spectrometers. To provide such a system is one of the major problems in the field of mass spectrometry.